6.4. The Modified Methods ofCharacteristics
181
verse method with a self-adaptive moving point technique. Their basic idea is
that the single step reverse method alone is used in the portion where the
front is smooth, while moving points are added automatically around the
steep front. When the front becomes smooth, the moving points will disappear automatically. The purpose of this method is to take advantage of the
simplicity of the single step reverse method and the effectiveness of the mo ving point method in defining the shape of steep front. This makes the presented
hybrid method applicable for both large and small Peclet number problems.
In general, for each transient state, the steep front only appears in a small
portion of the flow region. Let us first design a method to automatically
control the distribution of moving points so that they can be concentrated
around the steep front.
At time t k , define
SL = (Cf+l,j - Cf_l,j)2 + (CL+l - CL_d 2 ,
(6.4.11)
and prescribe a critical value S*. When SL < S*, the front in element (i,j) is
smooth, otherwise, it is steep. Furthermore, let the allowed minimum and
maximum numbers of moving points in the element with a smooth front be
mb' Mb' respectively. Similarly, define ms and Ms for the element with a sharp
front. The number of moving points, Vi~j' can be controlled automatically by
the following rules:
1. mb ~ Vi~j ~ Mb, when stj < S*;
2. ms ~ Vi~j ~ M s' when SL > S*.
If the number of moving points in an element exceeds the requirements of
the above rules, some moving points should be eliminated; otherwise, if they
are insufficient, new points need to be generated. This process can be controlled automatically by the computer pro gram.
In using the hybrid method, we first form the initial concentrations of all
nodes {C~j} according to the initial conditions, then compute {S~j}' If S~j <
S*, m b moving points should be placed in the element; otherwise, m s moving
points should be placed. The concentrations of the moving points {C~} are
determined based on the initial conditions, where the subscript v represents
the moving point. If concentrations {CL}, moving points {v k }, and the concentrations of these points {C!} have been obtained for time t k , we can use
the following steps to compute {ct? }, {Vk+l }, and {C!+l }:
1. Apply the single step reverse method to solve for the concentrations of all
nodes at tk+l' which are written as {ct?}. The dispersion part, i.e., the
term within the square brackets on the right-hand side of Eq. (6.4.9), is
written as <5CL;
2. Use interpolation to estimate the effect of dispersion on the concentrations of the moving points, {<5C!}, and modify the moving point concentrations as folIo ws
(6.4.12)
181
verse method with a self-adaptive moving point technique. Their basic idea is
that the single step reverse method alone is used in the portion where the
front is smooth, while moving points are added automatically around the
steep front. When the front becomes smooth, the moving points will disappear automatically. The purpose of this method is to take advantage of the
simplicity of the single step reverse method and the effectiveness of the mo ving point method in defining the shape of steep front. This makes the presented
hybrid method applicable for both large and small Peclet number problems.
In general, for each transient state, the steep front only appears in a small
portion of the flow region. Let us first design a method to automatically
control the distribution of moving points so that they can be concentrated
around the steep front.
At time t k , define
SL = (Cf+l,j - Cf_l,j)2 + (CL+l - CL_d 2 ,
(6.4.11)
and prescribe a critical value S*. When SL < S*, the front in element (i,j) is
smooth, otherwise, it is steep. Furthermore, let the allowed minimum and
maximum numbers of moving points in the element with a smooth front be
mb' Mb' respectively. Similarly, define ms and Ms for the element with a sharp
front. The number of moving points, Vi~j' can be controlled automatically by
the following rules:
1. mb ~ Vi~j ~ Mb, when stj < S*;
2. ms ~ Vi~j ~ M s' when SL > S*.
If the number of moving points in an element exceeds the requirements of
the above rules, some moving points should be eliminated; otherwise, if they
are insufficient, new points need to be generated. This process can be controlled automatically by the computer pro gram.
In using the hybrid method, we first form the initial concentrations of all
nodes {C~j} according to the initial conditions, then compute {S~j}' If S~j <
S*, m b moving points should be placed in the element; otherwise, m s moving
points should be placed. The concentrations of the moving points {C~} are
determined based on the initial conditions, where the subscript v represents
the moving point. If concentrations {CL}, moving points {v k }, and the concentrations of these points {C!} have been obtained for time t k , we can use
the following steps to compute {ct? }, {Vk+l }, and {C!+l }:
1. Apply the single step reverse method to solve for the concentrations of all
nodes at tk+l' which are written as {ct?}. The dispersion part, i.e., the
term within the square brackets on the right-hand side of Eq. (6.4.9), is
written as <5CL;
2. Use interpolation to estimate the effect of dispersion on the concentrations of the moving points, {<5C!}, and modify the moving point concentrations as folIo ws
(6.4.12)